Constants External Anchors Hbar Si Pos

The reduced Planck constant

The reduced Planck constant, written ħ and pronounced h-bar, is the quantum of angular momentum. It is the Planck constant h divided by 2π, and it sets the scale at which quantum effects become unavoidable: a particle's angular momentum comes in whole multiples of ħ, and the energy of a photon of frequency f is h f, equivalently ħ times its angular frequency. In the International System of Units, its value is defined exactly as 1.054571817 × 10⁻³⁴ joule-seconds, a definition fixed by the 2019 SI redefinition rather than measured anew.

The framework Recognition Science (RS) keeps a machine-checked library of formal theorems, and inside that library a single module, ExternalAnchors, holds every empirical calibration value that enters RS from outside. The module records ħ in SI units as the number 1.054571817e-34, and a small lemma named hbar_SI_pos proves the trivial fact that this number is greater than zero. The lemma is a formal statement about the sign of a defined constant, not a measurement and not a derivation of the constant's value.

The purpose of the module is mechanical separation. The cost-first core of RS, the part that derives structure from the recognition ledger, never imports ExternalAnchors. Any module that does import it is explicitly acknowledging that it uses external calibration data. This quarantine means the derived constants of RS, such as ħ = φ⁻⁵ in the framework's own units, are never entangled with the empirical values used to compare against experiment. The positivity lemma exists so that downstream proofs can rely on ħ being a positive real number without re-deriving that fact each time.

What hbar_SI_pos does not claim is as important as what it proves. It does not assert that RS derived the value of ħ from first principles; the value is copied from CODATA 2022. It does not assert that the framework's native ħ, expressed in terms of the golden ratio, equals the SI value; the relationship between the two unit systems is a separate matter. And the lemma says nothing about the physical meaning of ħ, only that the recorded number is positive.

/-- **EXTERNAL ANCHOR**: Reduced Planck constant (exact, SI 2019 definition).
    ℏ = 1.054571817... × 10⁻³⁴ J·s -/
@[simp]
noncomputable def hbar_SI : ℝ := 1.054571817e-34

lemma hbar_SI_pos : 0 < hbar_SI := by norm_num [hbar_SI]

/-- **EXTERNAL ANCHOR** marker type for documentation.
    This doesn't affect code but signals calibration dependence. -/
abbrev ExternalAnchorMarker := Unit

What this page does not claim

The lemma does not derive the value of ħ from the cost-first core; the value is an external calibration datum. The lemma does not assert equality between the framework's native ħ and the SI value. The lemma says nothing about the physical interpretation of ħ beyond its recorded numerical positivity.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Constants/ExternalAnchors.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: